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- "Michael Kac", Re: 2.595 Is Language Infinite?
- Tom Lai, Is language infinite?
- Terry Langendoen, is language infinite?

Answer to Linguist subscriber Macrakis on how Langendoen and Postal get uncountably infinite languages: the answer is that they argue, contrary to the standard position, that there are sentences of infinite length. (See Chapter 3 of their book.) Alexis Manaster-Ramer makes a couple of comments in his most recent posting that I want to strongly second. The first is that the question of finiteness/ nonfiniteness of languages is not a factual one; the second is his expression of discomfort with the cavalier way in which the Langendoen-Postal thesis is rejected by most linguists. I'm reluctant to take up a lot more space with my endless 2 cents worth but encourage Linguist subscribers who are interested in the issue to correspond with me personally. My address is kaccs.umnedu. Michael KacMail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue

I appreciate Alexis Manaster Ramer's remarks of Sept. 29. I apologize for any previous strong-worded comments of mine. The notions of countable and uncountable infinity are, I agree, mathematical convenience. But, that they are such only because they are useful, i.e. they do reflect something that is significant. For example, a properly written computer program should, at least, terminate after a _finite_ number of operations. If there is an unending loop like WHILE TRUE DO BEGIN STATEMENT END then the number of operations will be _(countably) infinite_. The physical significance is that this program will never terminate. As for the cardinality of language or of individual languages, natural or not natural, I have been silent for some time to sort out my ideas and to listen to other contributors to the discussion. Now, I would say (1) To use computing jargon, any (particular) language over a finite alphabet with no upper bound on the length of its strings is _countably_finite; (2) Natural languages have finite inventories of phonemes. It follows from (1) that if there is no upper bound on the length of words (there may be dispute over this point), the lexicons can be (countably) infinite; (cf. linguists' perception of "open" lexicons) (3) If the sentences of a language is to be formed by words extracted from an infinite (countable) lexicon, and as recursion entails the possibility of sentences of any length, the set of sentences (of this language) is _uncountably_ infinite. Point (3) above should answer Macrakis' query on the possibility of uncountable cardinality in set theory. We will not get into any paradox (a concern of Macrakis'). Whereas natural languages can be perceived as infinite, any actual simulation of a natural language on a computer is finite. (Can I assume that it is impossible for us to have infinite computer lexicons?) Sorry for the length of this posting. Looking forward to views of samretanos. Tom Lai.Mail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue

In a recent LINGUIST posting, Alexis Manaster-Ramer writes: The same applies, of course, the Langendoen/Postal work: I still cannot believe that Terry and Paul could seriously claim that the issue of whether NL sentences are merely of unbounded length or of infinite length (and whether the collection of English sentences was countable or not) a factual question. But by the same token I cannot understand the self-righteous dismissal of their proposals by those who somehow possess the certitude (that I so notably lack) that NL sentences are only of finite length. I cannot speak for Paul, but I agree with Alexis that the question of the size of natural languages is a theoretical one. As a matter of fact, we can all agree (even Jacques Guy), that NLs are at least of some finite size. How much larger they are depends on whether one considers that grammars of natural languages contain what Paul and I called "size laws", of which the following are possibilities. (For simplicity, I assume that these all deal with the notion of "length".) 1. There is some fixed, finite n, such that all sentences of all natural languages are no longer than n. Something like this assumption was made by Peter Reich in a paper in LANGUAGE in 1969. 2. All sentences of all natural languages are of finite length, but there is no n, as in (1). This is the standard Chomskyan assumption. 3. There are no size laws. That is there are no priniciples of grammar that either explicitly or implicitly limit the length of sentences in natural languages. This is the assumption that Paul and I defend, basically on simplicity grounds. Paul and I show that given (3), and given a principle of grammar concerning coordinate compounding, there are transfinitely many sentences in all natural languages for which that principle holds. The principle itself is quite simple and we think uncontroversial. It says in effect if expressions of type X (such as the type of declarative sentences) are compoundable by coordination, then for any two or more expressions of that type, there is at least one coordinate compound in the language, also of type X, made up of those expressions. To see how this works in a simple case, let us limit our (uncompounded) expressions of type X to the countably infinite set B = <<(I know that)**n Babar is happy: n >= 0>>. (Note: I use << and >> as set delimiters, to avoid transmission difficulties with the curly brace symbols.) From the principle, there is an expression of type X for every member of the power set of B, excluding the empty set. Call this set B'. For example, B' contains <<Babar is happy, I know that I know that Babar is happy>>, and corresponding to this member is the expression, also of the appropriate type, Babar is happy and I know that I know that Babar is happy. (In fact, there are other expressions as well, but we ignore these.) By standard assumptions of set theory, B' has greater cardinality than B. Paul's and my result then follows, unless one imposes a size law such as (2), which effectively (and in our view arbitrarily) limits the application of the coordinate compounding principle to sets of expressions of finite cardinality (that is, one replaces the "two or more" in my formulation above by the range 1 < x < infinity, where x ranges over the number of expressions that are coordinately compounded). Paul and I argue that the best theories of natural languages are those which do not countenance size laws, which we claim have two properties: 1. they are not constructive (e.g., generative); 2. they support a realist (i.e., Platonist) view of natural languages. The first property certainly must hold, but given the spirit of much contemporary theorizing is less controversial than it was when we published our book in 1984, if it was even controversial then, given the rise of "principles and parameters" based theories of natural languages during the 1980s. The second property, it now strikes me, does not hold. Paul and I spilled a lot of ink trying to clarify Chomsky's "conceptualist" view of natural languages, and having clarified it, to show that it is incorrect. But it now strikes me as quite easy to work out a coherent conceptualist theory of natural languages that does not incorporate a size law. In fact, any principles and parameters theory that does not do so, but that does include what strikes me as the patently correct principle of coordinate compounding, will yield the result that natural languages (albeit, E-languages, in Chomsky's terminology) are of transfinite size.Mail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue